Damped Oscillator

Damped Oscillator

Real oscillators lose energy to friction and air resistance. The damped oscillator adds a velocity-proportional damping term:

$x'' + 2\zeta\omega\, x' + \omega^2 x = 0$

• $\omega$: natural frequency (rad/s)
• $\zeta$ (zeta): damping ratio (dimensionless)

Three Regimes

As a System

Let $v = x'$:

$\frac{dx}{dt} = v$ $\frac{dv}{dt} = -2\zeta\omega\, v - \omega^2 x$

Numerical Solution (Symplectic Euler)

a = -2*zeta*omega*v - omega**2 * x
v = v + h * a
x = x + h * v

Applications

• Suspension systems in cars (want $\zeta \approx 0.7$ for comfortable ride)
• Building dampers for earthquake resistance
• RLC circuits in electronics
• Atomic force microscopy cantilevers

Your Task

Implement damped_oscillator(omega, zeta, x0, v0, t_end, n). Return (x, v).